This section is intended to introduce various aspects of the art, which may be associated with embodiments of the invention. A list of references is provided at the end of this section and will be referred to hereinafter. This discussion, including the references, is believed to assist in providing a framework to facilitate a better understanding of particular aspects of the invention. Accordingly, this section should be read in this light, and not necessarily as admissions of prior art.
Computer models of erosion, transport and deposition of sediments by both water flow and turbidity current are important tools in a wide array of environmental, engineering and energy industries. These models are needed for designing bridge piers, dredging channels and harbors, and protecting beaches and wetlands. Recently, these models have also been applied to build geologic models for exploration, development and production of energy sources in oil and gas industries.
A geologic model is a digital representation of the detailed internal geometry and rock properties of a subsurface earth volume, such as a petroleum reservoir or a sediment-filled basin. In the oil and gas industry, geologic models provide geologic input to reservoir performance simulations which are used to select locations for new wells, estimate hydrocarbon reserves, and plan reservoir-development strategies. The spatial distribution of permeability is a key parameter in characterizing reservoir performances, and, together with other rock and fluid properties, determines the producibility of the reservoir.
The spatial distributions of permeability in most oil and gas reservoirs are highly heterogeneous. One of the principal causes of the heterogeneity is the distribution of different grain sizes of sediments in different places in the reservoir. This is because most clastic reservoirs were formed by the deposition of the sediments in ancient fluvial, deltaic and deep water depositional systems. Since sediment grains with different sizes are eroded and transported differently, they are then been deposited in different places in the reservoir. Therefore if erosion, transport and deposition of the sediments can be accurately modeled in depositional systems, the heterogeneities of reservoirs formed by these depositional processes can also be accurately captured.
Two steps used to develop a model for sediment erosion, transport and deposition are 1) establishing a re-suspension (also known as erosion or entrainment) relationship between different size sediment grains and flows of different strength, and 2) characterizing the distribution of the suspended sediments in the vertical direction of the water column in which the sediments are suspended. In a single or multiple layer depth averaged flow model, the vertical distribution of the suspended sediments is described by the relationship that relates the near bed sediment concentration to the depth average concentration for different sizes of the grains. The second step is important because the deposition of the suspended sediments from the flow to the bed is closely related to the concentration of the sediments suspended just above the bed. In single or multiple layer depth averaged flow models, only the layer averaged values of the sediment concentration can be computed. These layer averaged concentration values can be significantly different from the actual concentration values that are just above the bed. Therefore the relationship that relates the depth averaged concentration to the near bed concentration is a necessary step for accurate calculations of the deposition of the sediments and subsequently, modeling of the sediment transport.
The most commonly used re-suspension (erosion) relationship is known as the Garcia-Parker re-suspension function. In this function, the rate of re-suspension of sediments of grain size bin i into the flow from the bed is
                                                                        E                i                            =                            ⁢                                                E                                      s                    i                                                  ⁢                                  v                  si                                ⁢                                  G                  i                                                                                                        =                            ⁢                                                                                          a                      z                                        ⁢                                          Z                      i                      5                                                                            1                    +                                                                                            a                          z                                                                          e                          m                                                                    ⁢                                              Z                        i                        5                                                                                            ⁢                                  v                  si                                ⁢                                  G                  i                                                                                        [        1        ]            where Esi is the dimensionless re-suspension rate and is related to the dimensional re-suspension rate by
                              E          si                =                                            E              i                                                      v                si                            ⁢                              G                i                                              =                                                                      a                  z                                ⁢                                  Z                  i                  5                                                            1                +                                                                            a                      z                                                              e                      m                                                        ⁢                                      Z                    i                    5                                                                        .                                              [        2        ]            
In Equation [1], Gi is the volumetric percentage of the sediments of grain size bin i in the surface layer, vsi is the settling velocity for a sediment grain with diameter Di in the i th size bin, az is a constant and typically has a value of 1.3×10−7, and em equals the maximum value of the dimensionless re-suspension rate Esi. The value of em sets the upper limit on the value of the re-suspension function.
Function Zi shown in Equation [1] is defined as
                              Z          i                =                  λ          ⁢                                    u              *                                      v              si                                ⁢                      f            ⁡                          (                              R                pi                            )                                ⁢                                    (                                                D                  i                                                  D                  50                                            )                        0.2                                              [        3        ]            in whichλ=1−0.288σΦ.  [4]
In the above equations, u* is the shear flow velocity, D50 is the diameter of the sediment grain in the 50th percentile in the distribution, and σΦ is the standard deviation of the grain size distribution in the logarithmic “phi” units familiar to geologists. The particle Reynolds number Rpi for grains in the ith size bin is defined as
                              R          pi                =                                                            (                                  RgD                  i                                )                                            1                /                2                                      ⁢                          D              i                                v                                    [        5        ]            where R is the submerged specific weight of the sediments, g is the gravitational acceleration constant, and v is the kinematic viscosity of the water.
There are two commonly used formulations of the Reynolds function ƒ(Rpi) as used in Equation [3]. The first isƒ(Rpi)=Rpi0.6.  [6]
The second form of the Reynolds function adds a correction for particles with a small particle Reynolds number as follows:
                              f          ⁡                      (                          R              pi                        )                          =                  {                                                                      R                  pi                  0.6                                                                                                  if                    ⁢                                                                                  ⁢                                          R                      pi                                                        >                  2.36                                                                                                      0.586                  ⁢                                      R                    pi                    1.23                                                                                                                    if                    ⁢                                                                                  ⁢                                          R                      pi                                                        ≤                                      2.36                    .                                                                                                          [        7        ]            
Both forms of ƒ(Rpi) are in use. In this document, the re-suspension relationship defined by Equations [1], [3] and using the form of ƒ(Rpi) in Equation [6], is referred to as the Garcia Thesis Model. Correspondingly, the re-suspension relationship using the form of ƒ(Rpi) in Equation [7] is referred to as Garcia 1993 Model.
For sediments with a single grain size, the re-suspension relationship [3] shown above can be simplified because the terms λ and
            (                        D          i                          D          50                    )        0.2    ,both characterizing interactions among sediment particles of different grain sizes, are both reduced to 1. The function Zi can therefore be stated as
                              Z          i                =                                            u              *                                      v              si                                ⁢                                    f              ⁡                              (                                  R                  pi                                )                                      .                                              [        8        ]            
The re-suspension relationship using Equation [8] is often used to study flume experiments.
Another example of a re-suspension function that could be used is from Akiyama and Fukushima. In this re-suspension function, the rate of re-suspension of sediments of the grain size bin i into the flow is:
                              E          i                =                  {                                                                                          0.3                    ⁢                                                                                  ⁢                                          v                      si                                        ⁢                                          G                      i                                                        ,                                                                                                  Z                    i                                    >                                      Z                    m                                                                                                                                            3                    ×                                          10                                              -                        12                                                              ⁢                                                                  Z                        i                        10                                            ⁡                                              (                                                  1                          -                                                                                    Z                              c                                                                                      Z                              i                                                                                                      )                                                              ⁢                                          v                      si                                        ⁢                                          G                      i                                                        ,                                                                                                  Z                    c                                    ≤                                      Z                    i                                    ≤                                      Z                    m                                                                                                                        0                  ,                                                                                                  Z                    i                                    <                                      Z                    c                                                                                                          [        9        ]            where Zc=5 and Zm=13.2. The calculation of Zi is the same as that shown in Equations [3] or [8].
The re-suspension relationships shown in Equations [1] and [9] are necessary closure relationships in the calculations of transport of sediments in natural conditions, using any flow models. Examples of these flow models include depth-averaged flow models, full 2D flow models, and full 3D flow models. Here the full 2D and 3D flow models refer to models where the variations of the flow properties and sediment concentrations in the vertical direction are variables of the governing flow equations. Many flow models used in existing commercial software such as Fluent or Flow3D are of this class.
Another important closure relationship that is necessary when the depth-averaged flow models are used is the relationship between r0 (which represents the ratio of the near-bed depth to the depth averaged sediment concentrations) and the flow and sediment conditions. When sediment is transported by turbulent flow, the distribution of the sediments in the vertical direction is not uniform, but instead certain sediment concentration profiles are formed. Usually the concentration of the sediments is greater in the bottom part of the flow (i.e., nearest the bed) than in the upper part of the flow. Near-bed sediment concentrations refers to the sediment concentrations at the place that is right above the bed. The actual location is often treated as a model parameter. Typical values range from a distance equal to the diameter of the largest grain on the bed, to 10% of the flow depth.
The most commonly used expression for r0 is the constant approximation wherer0=const  [10]in which the constant typically has a value range between 1.0 to 2.5. Another expression of r0 isr0=1+31.5μ−1.46  [11]in which
                    μ        =                              u            *                                v            s                                              [        12        ]            is the ratio between the shear velocity u* and the particle fall velocity vs. In a mixture of sediments with multiple size particles, r0i can be substituted in the above equations for r0, and vsi for vs, corresponding to sediment grains in the size bin i, respectively. Other expressions of r0 include a linear formr0=2.0761−0.0108μ  [13]and a simple power law formr0=2.2461μ−0.0772.  [14]Similar to Equations [10] and [11], r0i is substituted for r0 and vsi for vs for sediment mixtures of multiple size particles.
Both the re-suspension functions and the near bed to depth averaged sediment concentration relationships shown in Equations [1], [9], [10], [11], [13] and [14] were mostly obtained from flume experiments using sediment mixtures with very fine mean grain sizes and with narrow grain size distributions. While these equations are suitable in such circumstances, important deficiencies and significant inconsistencies are observed in all of the equations when applied in certain real-world conditions. These deficiencies and inconsistencies will now be discussed in more detail.
Since the dimensionless entrainment rate Esi shown in Equation [2] is a monotonic function of Zi, a threshold value Zt can then be chosen to define the onset of significant suspension. According to Equation [3] or [8], Zi is a function of the shear flow velocity u* and the grain size Di. Therefore curves Zi(u*,Di)=Zt can be drawn in a u*-D plot to indicate the location of the onset of the significant suspension in u*-D space for any choice of Zt.
FIG. 1A shows plots 11, 12 and 13 of the curves Zi(u*,Di)=Zt obtained using the Garcia Thesis Model and corresponding to three different choices of Zt, namely 1, 5 and 10, respectively. FIG. 1B shows curves 21, 22 and 23 obtained using the Garcia 1993 Model and corresponding to the same values of Zt. For comparison, the Shields curve 14, which describes the critical shear velocity needed for the beginning of motion of particles of size Di on the bed, has also been plotted in FIGS. 1A and 1B. The Shields curve was developed from flume experiments and is used in many bedload transport relationships. For further comparison, the curve 15 of u*=vs(Di) has also been plotted in FIGS. 1A and 1B. Curve 15 shows when the shear velocity has the same value as the falling velocity of the particle of the size Di. In most situations, significant suspension can not occur when the shear flow velocity u*<vs(Di). Therefore, this curve provides a good lower bound for the beginning of suspension.
As expected, FIGS. 1A and 1B show that the curves plotted with different values of Zt=1, Zt=5 and Zt=10, are different. Larger values of Zt imply a higher threshold for the onset of suspension, and consequently, correspond to higher threshold values of u* for the same D. A commonly used value is Zt=5.
It can be seen immediately from FIGS. 1A and 1B that results for the onset of significant suspension obtained using both the Garcia Thesis Model and the Garcia 1993 Model are incorrect for sediments of grain sizes greater than about 1.5 mm. Specifically, the curves predict that shear flow velocity u* will decrease as grain sizes D increase above about 1.5 mm. This is contrary to common knowledge that larger particles are heavier and more difficult to move, and thus are less likely to be suspended than lighter particles. The results shown in FIG. 1A, however, incorrectly suggest that a larger flow velocity is required to suspend a particle with diameter 1.5 mm than to suspend a particle with diameter of 100 mm.
A second inconsistency of previous re-suspension models is that the curves for the onset of significant suspension plotted in FIGS. 1A and 1B drop far below the u*=vs curve and the Shields curve for large grain size sediments. It has been observed that most natural rivers characterized by a dominating suspended sediment load are plotted above the u*=vs curve. The natural rivers that are plotted between the u*=vs curve and the Shields curve are mostly bed load dominated, and the suspension of sediments therein is not significant. As curves 11-13 and 21-23 fall even below the Shields curve for large grain size sediments, the re-suspension models used to generate curves 11-13 and 21-23 do not accurately predict behavior of large grain size sediments.
The same inconsistency becomes clearer when the curves for the onset of significant suspension are used to predict the shear flow velocity u* for sediments with grain sizes greater than, for example, 4 mm, if the Zt=5 curve 12, 22 is used as the criteria. In this case, the Zt=5 curve 12, 22 drops below the Shields curve 14 for all the sediments having D>4 mm, which implies that a significant amount of suspension can occur at a value of u* that is smaller than the critical value of u* for any bed load to occur. Once again, the known functions relating to the onset of significant suspension, as plotted in FIGS. 1A and 1B, are clearly incorrect.
The curves for the onset of significant suspension corresponding to the Garcia 1993 Model (FIG. 1B) differ from that of the Garcia Thesis Model (FIG. 1A) in that a correction for sediment particles with small particle Reynolds numbers has been added. This correction mitigates the error in the Garcia Thesis model where the curves for suspension onset drop below the Shields curve for very fine sediments. For example, the curve 12 corresponding to Z=5 in FIG. 1A drops below the Shields curve 14 when the sediment grain size is less than about 0.05 mm. Although the corresponding Z=5 curve 22 in FIG. 1B also drops below the Shields curve 14 for very small sediment grain size, such crossover occurs when the grain size decreases to less than about 0.008 mm. Therefore, despite the improvement the Garcia 1993 Model made with regard to the original Garcia Thesis Model, the segment of each curve that corresponds to small particles is still convex, as is the case with FIG. 1A. This convexity of the curves is also manifest in the portions of the curves corresponding to large particles. The convex nature of the curves predicts an increasing better sorting of the sediments with decreasing mean grain sizes of D<0.07 mm, which is inconsistent with field observations. In this context, sorting refers to how a fluid flow deposits sediments. It is assumed that larger, heavier sediments are deposited before smaller, lighter sediments.
The convex nature of the small particle Reynolds number segment of curves 11-13, 21-23 also results in near zero slope in these curves for sediments with grain sizes from 0.02 mm to 0.07 mm. This implies that for sediments of single grain sizes in this grain size range, the re-suspension rates of the sediments does not significantly vary. Under such conditions, however, the Garcia 1993 Model could predict a reversed sorting (i.e., the fluid flow deposits smaller sediments before larger sediments) if the terms for the interactions among sediments of different grain sizes shown in Equation [3] are also taken into account. In FIG. 2, the dimensionless sediment re-suspension rate ES is plotted as a function of the shear flow velocity for different grain sizes in the mixture as follows: a grain size of 6.25 microns is shown by the lighter solid line 25, a grain size of 12.5 microns is shown by the dotted line 26, a grain size of 25 microns is shown by the dashed line 27, a grain size of 50 microns is shown by the darker solid line 28, a grain size of 100 microns is shown by the plotted circles 29, a grain size of 200 microns is shown by the plotted squares 30, a grain size of 400 microns is shown by the plotted+signs 31, and a grain size of 800 microns is shown by the plotted triangles 32. Equations [3] and [7] (i.e., the Garcia 1993 Model) are used to obtain the results shown in the Figure. It is clear from FIG. 2 that the dimensionless re-suspension rate ES decreases as the sediment grain size increases, except when the grain size is 50 microns. Contrary to what would be expected, the dimensionless re-suspension rate for sediments with grain size of 50 microns is greater than the re-suspension rate for sediments with smaller grain sizes of 25 microns and 12.5 microns. The results predicted by the Garcia 1993 Model are therefore incorrect.
Similar to the inconsistencies of the existing re-suspension functions as outlined above, the known functions expressing the ratio between the near-bed sediment concentrations and the depth averaged values also have many significant deficiencies. FIG. 3 shows the ratio r0 between the near-bed sediment concentration and the depth averaged concentration, as a function of μ, where
  μ  =                    u        *                    v        s              .  Two sets of experimental data, identified in the Figure as the Garcia data set 34 (diamonds) and the Graf data set 35 (circles), are plotted in the Figure. Also shown in the Figure are various approximations for r0, such as a constant approximation 36 as suggested by Equation [10], a simple linear fit 37 as suggested by Equation [13], a simple power law approximation 38 as suggested by Equation [14], and an approximation 39 as calculated from Equations [11] and [12]. The values for r0 according to curves 36, 37 and 38 are not good fits for the experimental data 34, 35, especially for small values of μ (i.e., less than 10). For example, when μ→0, the value of r0 is about 2.1 and 3.1 according to the simple linear fit 37 and the simple power law approximation, respectively. Note that μ→0 corresponds to situations when there is little turbulence in the flow and sediments get little lift to stay suspended in the flow. In these situations, most of the sediments will be concentrated in the very bottom part of the flow. The value of r0 is therefore expected to be significantly greater than the numbers predicted by these two approximations.
The approximation 39 appears to be a somewhat better fit to the experimental data 34, 35 but is still seriously deficient for very small values of μ. Specifically, in the approximation 39 r0→∞ when μ→0. This is not correct. Let Cb be the near-bed sediment concentration and C be the depth averaged sediment concentration. The near-bed sediment concentration Cb is defined as the sediment concentration measured at the distance from the bed equal to certain fraction δ of the flow height. In the limiting case when all the sediments are below the level of δh, where h is the flow height, Cbδh<Ch. Therefore
      r    0    =                    C        b            C        <                  1        δ            .      In practice, a value of 0.05 is often used for δ. In that case, the upper bound for r0 is 20.
The re-suspension relationship defined in Equations [1] and [3] is a continuous function with regard to u*. When this form of re-suspension relationship is used together with bed load transport equations, which often contain cut-off thresholds based on critical shear stresses, inconsistencies could arise. FIG. 4 shows the comparison of the volumetric transport per unit width q between the bed load transport, represented by curve 40 and the suspended load transport, represented by curve 41. The bed load transport relationship used in the calculation is taken from the Ashida reference cited herein. The suspended load is calculated using the Garcia 1993 Model given in Equations [1], [3], [4], [5] and [7]. The calculation used twelve bins of sediments with a minimum grain size of 6.25 micron and a maximum grain size of 1.280 cm and assumed a log-uniform distribution. The results shown in FIG. 4 correspond to the sediments with a grain size of 1.6 mm. For shear flow velocity u*>0.35, FIG. 4 shows that the sediment transport was initially dominated by the bed load. As u* increases, the suspended load becomes the main component of the sediment transport, as expected. It can also be seen from the figure that for u*<0.29, the bed load drops to zero because the critical shear flow velocity is not exceeded. However, different from the bed load, since Es is a continuous function of u*, the suspended load is not zero. Although the rate of the suspended load transport must also be small, the results of FIG. 4 nonetheless imply the suspension of 1.6 mm size sediments prior to the occurring of any bed load transport, which cannot be correct.
Many problems and inconsistencies with existing theories of re-suspension of sediments in turbulent flows have been set forth herein. A model that eliminates these problems and inconsistencies is needed. The present invention provides such a model.
Other related material may be found in the following: U.S. Pat. No. 70,201,300; Akiyama, J., and Fukushima, Y. (1986), Entrainment of noncohesive sediment into suspension, 3rd Int. Symp. on River sedimentation, S. Y. Wang, H. W. Shen and L. Z. Ding, eds., Univ. of Mississippi, 804-813; Garcia, Ph.D thesis, University of Minnesota, 1989 [Inventors: need full cite here]; Garcia and Parker, Entrainment of bed sediment into suspension, Journal of Hydraulic Engineering, 117(4), pp 414-435, 1991; Garcia and Parker, Experiments on the entrainment of sediment into suspension by a dense bottom current, Journal of Geophysical Research, 98(C3), 4793-4807, 1993; Garcia, M. H. (1999), Sedimentation and erosion hydraulics, Hydraulic design handbook, L. Mays, ed., McGraw-Hill, New York, 6.1-6.113; Graf, W. H., (1971), Hydraulics of sediment transport. McGraw-Hill Book Co., Inc., New York, N.Y. [Inventors: need page numbers here.]; Parker, G., Fukushima, Y., and Pantin, H. M. (1986), Self-accelerating turbidity currents, J. Fluid Mech., v 171, 145-181; Ashida, K. and Michiue, M. (1971), An investigation of river bed degradation downstream of a dam, Proc. 14th Congress of the IAHR. [inventors: need full cite]; and Garcia, M. H., Depositional turbidity currents laden with poorly sorted sediment, Journal of Hydraulic Engineering, v 120, No. 11, pp 1240, (1993).